Abstract:
Fractals describe the increase in the length of a line with a
decrease in the scale at which it is measured. A greater fractal
dimension at a particular scale may translate into more habitat
space for an animal of that size, and therefore greater expected
abundances of that animal. I tested for the presence of such a
relationship between the alga Mastocarpus papillatus and the small
snails, Littorina scutulata and Tricolia pulloides, living on it.
The use of M. papillatus provided a range of morphologies so that
the effects of fractal dimension could be examined within a species
of alga. This intra-species comparison allowed for a control of
the grazing quality of the alga. Indeed higher fractal dimension
was shown to correlate with a higher density of snails.
To determine if a fixed number of snails is associated with
each thallus, experiments were performed in which the natural
populations of snails on M. papillatus thalli were altered. The
number of snails on each thallus was found to return to the
original number of snails the day after treatment. This result
indicates that there is a relationship between snail density and an
individual alga.
Introduction:
Mandelbrot (1967, 1982) pioneered the use of fractals as an
alternative to euclidian geometry in describing the world. For a
fractal object, in contrast to a euclidian object, the perceived
length or area of that object increases as the unit of measurement
decreases. The rate of increase is determined by the fractal
dimension according to the equation L(s)-Ks (1-D) where L is the
length measured at a scale s, K is a constant, and D is the fractal
dimension. Fractal dimensions have been determined for habitat
space such as lichens (Shorrocks et al, 1991), vegetation (Morse et
al, 1985) and algae (Gee and Warwick, 1994). Fractals describe the
complexity of the substrate, and therefore how much habitat is
available at a given scale. The higher the fractal dimension
measured at a given scale, the higher the expected density of
animals of that body size that could be supported.
Gee and Warwick (1994) found a relationship between the
fractal dimensions of macroalgae and the diversity and structure of
their epifaunal populations. They measured the fractal dimensions
of 4 intertidal algae found on the Isles of Scilly, off of the
coast of Britain. They found a higher density of animals on plants
with higher fractal dimensions. However, as they noted, they only
used one frond to measure the fractal dimension of a species of
alga, and took that frond to be representative of all individuals.
They then used that dimension for their inter-species comparison.
The fractal dimension measured may have been non-representative of
the population of individuals sampled, and therefore could have
caused some bias in comparing fractal dimension to animal
abundance.
In this study I follow the lead of Gee and Warwick (1994) but
measure the fractal dimension of each individual alga sampled so
that the relationship between fractal dimension and animal
abundance can studied more closely. Studying the variation of
fractal dimension within one species also allows me to control for
the grazing quality the alga provides to its resident snails. By
controlling for grazing quality, wave exposure, and tidal height,
I hoped to determine how significant the fractal dimension of algae
is for the animals living on it.
Materials and Methods:
The intertidal red alga Mastocarpus papillatus was studied
because it exhibits a wide range of morphologies in the field and
occurs commonly along the Pacific Coast of North America (Bell,
1992). The snails Littorina scutulata and Tricolia pulloides were
used as indicators of animal abundance because they were the most
abundant animals on M. papillatus in my study site at Hopkins
Marine Station and because of the ease of counting and manipulating
them. The only other abundant macroscopic animals living on the M.
papillatus at Hopkins Marine Station were amphipods, and their high
mobility made them difficult to quantify and handle. It was
assumed that amphipod abundances did not affect snail abundances.
Forty-nine Mastocarpus papillatus were collected randomly from
an exposed site at Hopkins Marine Station, 39 from a tidal height
between 0 and 1 meter, the remainder from between 1 and 2 meters
Each alga was placed in a separate container and taken back to the
lab for analysis. The macroscopic animals were hand picked from
each specimen, and their numbers and sizes were recorded. Each
alga was blotted with paper towels and weighed to the nearest 0.O1
g. The number of 1 to 3 mm Littorina scutulata and Tricolia
pulloides inhabiting each alga were counted. Their combined
density was measured in terms of number of snails per 7 grams of
algae. Seven grams was chosen as the normalization mass because it
approximated the mean mass of the samples. Some samples taken were
much smaller than 7 grams, with a few as low as 0.8 grams. These
samples had be multiplied by a factor greater than 7 to determine
a normalized snail density. This multiplication enlarged any error
associated with sampling, and it was decided that snail densities
associated with samples weighing less than 1.75 grams were
unreliable. The analysis was carried out both with and without
these suspect samples.
To determine the fractal dimension of M. papillatus, a high
contrast, black and white slide was taken of each individual. The
alga was spread so that individual fronds would be visible, yet
were allowed to maintain their 3-dimensional form. This was done
so that the relative "bushiness" of the samples would be observed
in the slides. Some samples lay flat, while others crossed over
themselves in a more complex fashion.
These slides were projected onto paper, and an outline of the
projected alga traced. This tracing was digitized and fed into a
computer program which analyzed the image according to the grid
method used by Mandelbrot (1982), Sugihara and May (1990), Morse et
al (1985), and Shorrocks et al (1991). A square grid divided into
4 smaller squares is fit around the image of the trace. The number
of squares the image enters is counted. Then each small square is
divided into four and the number of squares containing part of the
outline are again counted. This process is repeated n times such
that the final number of squares will equal 2'n. The natural
logarithm of the number of squares entered at each level n is
plotted against the natural logarithm of the length of a side of
the smallest individual square. The negative slope of this log-log
plot is the fractal dimension of the image.
A centimeter scale was photographed with each alga so that the
tracing could be scaled to life size. An n of 4 or 5 was used for
analyzing M. papillatus, depending on the size of the tracing.
This number of levels gave a smallest square size between 2 and 2.5
mm, which most closely approximates the body size of the snails
being counted, and is therefore an estimate of the scale at which
this species experiences and uses its habitat.
The natural logarithm of the fractal dimension of each alga
collected at the lower tidal height was plotted against the natural
log of the density of snails between 1 and 3 mm in size. A plot
was made with and without samples weighing less than 1.75 grams.
Algae that were uninhabited by snails were left out of the analysis
to exclude algae that were potentially undesirable places to live.
A separate test was conducted to determine whether the fractal
dimensions differed between plants that did and did not have snails
living on them. A third comparison was made between the fractal
dimensions of the algae collected at the higher site and those
collected at the lower site.
Population manipulation experiments were conducted to
establish whether the number of snails living naturally on a given
algal thallus would return to its prior level after perturbation.
The number of snails living on each of twelve algae was counted and
the algae marked by hole-punching adjacent algae. The population
of snails was doubled on 4 thalli, eliminated on 4 thalli, and left
the same on the remaining 4 thalli. The populations of snails on
each alga were then monitored over the next 3 days.
Results:
The fractal dimension of inhabited algae collected at the low
tidal height ranged from 1.34 to 1.60. The higher the fractal
dimension of a particular alga, the more snails that are likely to
inhabit that alga (Figs. 1 and 2). When including all samples
collected, the R"2 for the regression is 0.169 (fig. 1), indicating
that 16.98 of the variation in snail population density can be
accounted for by fractal dimension. The slope of this regression
is significantly different from zero (p«0.05, student's t-test).
When samples of weights less than 1.75 grams are eliminated, the
2 for the regression is 0.550 (fig. 2), indicating that 558 of
snail densities can be accounted for by fractal dimension. The
slope of this regression is significantly different from zero
(p«0.001, student's t-test).
The mean fractal dimension of populated algae was 1.46 while
the mean fractal dimension of unpopulated algae was 1.43, and the
difference in fractal dimension was not significant.
The
difference between algae at the low tidal height (mean D-1.39) and
at the high tidal height (mean D-1.47) was significant only at the
0.1 level.
In the experiments where snail populations on algae were
manipulated, the control thalli remained at their original numbers
over the 3 days (fig. 3). The thalli from which all snails were
removed all returned to their original populations the first day
after the manipulation and oscillated around 1008 on the second and
third days (fig. 4). The thalli on which the population of snails
was doubled dropped back down to their original numbers on the
first day but then continued to drop on the second and third days
(fig. 5).
Discussion:
Gee and Warwick found a correlation between the density of
macrofauna and meiofauna living on four species of intertidal algae
and the fractal dimension of those algae. My research also found
a relationship between fractal dimension of intertidal algae and
animal abundance; fractal dimension is able to account for up to
558 of the variation in abundance of snails on M. papillatus. In
addition, I demonstrated that there is an approximately steady¬
state number of snails that occurs on each alga. Fractal dimension
can help explain those observed snail densities.
Why use fractals to describe habitat complexity? Surface area
per weight has been used to describe complexity (Hicks, 1977, 1980,
Russo 1990). However, surface area is difficult to measure for
complex forms such as algae and it cannot specify complexity for a
specific scale. The fractal dimension of any substrate can be
determined at the scale of interest, and therefore the complexity
of habitat that a particular sized animal experiences can be
quantified. This allows examination of the distribution of animals
of differing body sizes within a habitat.
Why is habitat complexity such a strong determinant of
population densities? It could be simply that a more complex
substrate will provide more space for inhabitants, as well as more
food, shelter, and water. An increasing fractal dimension can
increase more than just space, however. It can contribute to more
suitable microclimate conditions such as lower temperatures and
higher humidity (Johnson, 1973). Johnson found that not only does
algal complexity affect microclimate, but that amphipod abundances
respond to microclimate conditions.
Fractal dimension is clearly not the only factor that is
important in determining animal abundance. I showed that the
fractal dimensions of inhabited and uninhabited algae are not
significantly different at similar tidal heights and wave
exposures. Other factors must be causing some algae to be more
attractive to animals than others such as an alga's position on the
rock, which influences its exposure to sunlight, and the density of
the surrounding algae, which could provide a more moist
environment.
Fractal dimension is a way of describing habitat complexity
independent of substrate type. Studies could be done placing
artificial substrates of known fractal dimensions in the intertidal
and examining colonization to see if the resulting densities
resemble densities found on substrate naturally found in the
intertidal. This is a way of answering the question of the
importance of substrate type. To supplement this type of research,
comparisons could be made between different substrates of similar
fractal dimensions to see whether animal abundances are similar,
If abundances are similar, then an area of study may be opened
wherein different communities and even ecosystems can be compared
through their fractal dimensions.
A cause and effect relationship between fractal dimension and
animal abundance still has not been demonstrated. Maybe the
fractal dimension of algae or other substrates in the field can be
changed and the animal abundances observed to see if they respond
as expected to this change in fractal dimension. The difficulty of
a study like this is that it is impossible to change fractal
dimension without changing all of the factors that depend on
fractal dimension. Therefore it would be interesting to discover
what effect fractal dimension has on factors that comprise a
suitable habitat such as temperature, humidity, water retention.
The relationship I have found between the fractal dimension of
M. papillatus and its snail abundances is preliminary, and only
provides a glimpse of what might be shaping animal communities in
the intertidal zone. More samples need to be analyzed and more
relationships established between animal abundances and substrate
complexity as measured by fractal dimension before the true
descriptive power of fractals can be known.
ACKNOWLEDGEMENTS
I thank Mark Denny for being my advisor. It was wonderful to
learn from him and his unique way of looking at the world. Thanks
also to the members of the Denny lab for being friendly and
interested. I am indebted to Chris Patten for enabling me to
photograph numerous samples of algae. I would also like to give
thanks to Jim Watanabe whose expertise on the intertidal zone is
invaluable and finally, to our T.A. Ellen Freund.
References:
Bell, E.C., 1992. Consequences of Morphological Variation in an
Intertidal Macroalga: Physical Constraints on Growth and Survival
of Mastocarpus Papillatus Kutzing. A PhD Dissertation for the
Department of Biological Sciences, Stanford University,
Gee, J.M. & R.M. Warwick, 1994. Body-size Distribution in a Marine
Metazoan Community and the Fractal Dimensions of Macroalgae.
Exp. Mar. Biol. Ecol., Vol 178, pp. 247-259.
Gee, J.M. & R.M. Warwick, 1994. Metazoan Community Structure in
Relation to the Fractal Dimensions of Marine Macroalgae. Mar.
Ecol. Prog. Ser., Vol 103, pp. 141-150.
Hicks, G.R., 1977. Species Composition and Zoogeography of Marine
Phytal Harpacticoid Copepods From Cook Strait, and Their
Contribution to Total Phytal Meiofauna. J. Mar. Freshwat. Res.,
Vol 11, pp. 441-469.
Hicks, G.R., 1980. Structure of Phytal Harpacticoid Copepod
Assemblages and the Influence of Habitat Complexity and
Turbidity. J. Exp. Mar. Biol. Ecol., Vol 44, pp. 157-192.
10
Johnson, S.E. II, 1973. The Ecology of Oligochinus lighti J.L.
Barnard 1969, A Gammarid Amphipod From the High Rocky Intertidal
Region of Monterey Bay, California. A PhD Dissertation for the
Department of Biological Sciences, Stanford University.
Mandelbrot, B.B., 1967. How Long is the Coastline of Britain?,
Statistical Self-similarity and Fractal Dimension. Science, No.
156, pp.636-638.
Mandelbrot, B.B., 1982. The Fractal Geometry of Nature. W.H.
Freeman, New York.
Morse, D.R., J.H. Lawton, M.M. Dodson & M.H. Williamson, 1985.
Fractal Dimension of Vegetation and the Distribution of Arthropod
Body Length. Nature Lond., Vol. 314, pp. 731-733.
Russo, A.R., 1990.
The Role of Seaweed Complexity in Structuring
Hawaiian Epiphytal Amphipod Communities. Hydrobiologia, Vol. 94,
pp. 1-12.
Shorrocks, B., J. Marsters, I. Ward & P.J. Evennett, 1991. The
Fractal Dimensions of Lichens and the Distribution of Arthropod
Body Lengths. Funct. Ecol., Vol. 5, pp. 457-460.
Sugihara, G. & R.M. May, 1990. Applications of Fractals in
Ecology. TREE, Vol. 5, no. 3, pp 79-86.
11
2
r2 = 0.169
p «0.05

0.30
Figure 1



0.40
0.35
0.45
Ln Fractal Dimension
0.50
0.30


8

0.35
0.40
0.45
Ln Fractal Dimension
0.50
r2 = 0.550
p «0.001
Figure 2
200
100
Day After Treatment
Figure 3
200
100
Day After Treatment
Figure 4
300
200
100
Day After Treatment
Figure 5